2.1: Angle Relationships | Geometry Honors

How it works

Vertical angles are congruent. A linear pair sums to a straight angle. Complementary angles sum to a right angle, while supplementary angles sum to a straight angle.

\alpha+\beta=180^\circ\quad\text{for a linear pair}

Worked examples

Example 1

A linear pair measures $\displaystyle (3x+10)^\circ$ and $\displaystyle (2x+20)^\circ$ . Find $\displaystyle x$ .

$\displaystyle 5x+30=180$ , so $\displaystyle x=30$ . The angles are $\displaystyle 100^\circ$ and $\displaystyle 80^\circ$ .

Example 2

Vertical angles measure $\displaystyle (4x-5)^\circ$ and $\displaystyle (2x+35)^\circ$ . Find their measure.

$\displaystyle 4x-5=2x+35$ gives $\displaystyle x=20$ ; each measures $\displaystyle 75^\circ$ .

Problems

Problem 2.1.1

Two lines intersect. One angle measures $\displaystyle 65^\circ$ . Find the other three.

Two intersecting lines with one angle labeled 65 degrees. — The angle opposite a given angle is vertical; adjacent angles form linear pairs.

Show worked solution

The vertical angle is $\displaystyle 65^\circ$ . Each adjacent angle is $\displaystyle 180^\circ-65^\circ=115^\circ$ .

Two intersecting lines with one angle labeled 65 degrees. The solution adds labeled relationships. — Vertical angles are 65 degrees; each adjacent angle is 115 degrees.

Problem 2.1.2

Complementary angles measure $\displaystyle 2x$ and $\displaystyle 3x$ degrees. Find them.

Show worked solution

$\displaystyle 5x=90$ , so $\displaystyle x=18$ . The angles are $\displaystyle 36^\circ$ and $\displaystyle 54^\circ$ .

Problem 2.1.3

An angle is $\displaystyle 24^\circ$ less than its supplement. Find the angle.

Show worked solution

Let its measure be $\displaystyle x$ . Then $\displaystyle x=(180-x)-24$ , giving $\displaystyle 2x=156$ and $\displaystyle x=78^\circ$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.1.1

Problem 2.1.2

Problem 2.1.3

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